I am very serious about this. If you are just learning this material as an Optician, go back now. This will only be confusing. If you are an advanced student and want the way the Physicists do this, then read on.

For the physicist, a center of curvature that is to the right of the surface gives a positive radius and the surface is convex. A center of curvature that is to the left of the surface gives a negative radius, and the surface is concave.

On the lenses above, the first lens has a positive and a negative radius, and one convex and one concave surface. Likewise, the second lens. The third lens has two convex surfaces, both with positive radii. The Lensmakers formula always has a plus sign for the first surface power and always has a negative sign for the second surface power. Thus, using the signs for the radii, the first lens above has a plus radius and so the power is plus, and the second surface has a negative radius, but the fraction is subtracted, so the result is that you add the power of the second surface.

In the second lens above, the radius of the third surface is negative, and the radius of the fourth surface is positive; the end result is that both surfaces have a negative sign when the formula is computed, and the result is negative the sum of the surface powers.

In the third lens the radius of the fifth surface is positive as is the sixth surface; when the back surface power is subtracted from the front surface power the result is what we get the 'Opticians' way.

The formula now looks like this, where the sign of the radius is used:
                    n - 1        n - 1
       DN =  - 
                      r1             r2

There is another form of this equation which gives the same result, and is more likely to be found in physics books:
                                    1               1
       DN = (n - 1) (+  -  )
                                     r1             r2

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